Review of Digital Communication Theory

Transcription

1 Chapter 1 Review of Digital Communication Theory We start by reviewing several important concepts which will be needed in the following chapters. 1.1 Maximum likelihood receiver In this section, we assume that the communication channel is corrupted by an additive white Gaussian noise (AWGN) with two-sided power spectral density N = W/Hz. The transmitter sends a signal chosen from the set of M signals fs m (t)g M 1 m=. We further assume that all the M signals are time-limited to [;T], where T is called the symbol duration. The corresponding model for this communication system is shown in Figure 1.1. The received signal r(t) is given by r(t) =s m (t) +n(t); (1.1) s (t) m r(t)=s (t)+n(t) m ML Receiver decision n(t) Figure 1.1: AWGN channel communication model 1.1

2 for some m f; 1;:::; M 1g. In (1.1), n(t) denotes the AWGN process with power spectral density Φ n (f )=N = W/Hz. Our goal is to develop a receiver which observes the received signal r(t) and determines which one of the M signals is being sent based on maximizing the likelihood function. In order to proceed, we need to define what the likelihood function is. First, we try to represent the signals in a more convenient form. By employing the Gram-Schmidt procedure, we can construct a set of N (N» M) orthonormal functions fffi n (t)g N n=1 (all are timelimited to [;T]) which spans the signal space formed by fs m (t)g M 1 m=. We augment this set of functions by another set of orthonormal functions fffi n (t)g 1 n=n so that the augmented set fffi +1 n(t)g 1 n=1 forms an orthonormal basis for the space of suare-integrable functions. Employing this basis, any suare-integrable function can be represented by a vector whose elements are coordinates with respect to the basis functions fffi n (t)g 1 n=1. Based on this representation, we can rewrite (1.1) as r = sm + n; (1.) where r = [r 1 ;r ;:::;r N ;:::] T ; sm = [s m1 ;s m ;:::;s mn ;:::] T for m =; 1;:::;M 1; n = [n 1 ;n ;:::;n N ;:::] T ; are the vectors representing r(t), s m (t), andn(t) 1, respectively. The coordinates are, respectively, given by, for k =1; ;:::, Z 1 r k = r(t)ffi k(t)dt; (1.3) 1 Z 1 s mk = s m (t)ffi k (t)dt for m =; 1;:::;M 1; (1.4) 1 Z 1 n k = n(t)ffi k (t)dt: (1.5) 1 1 We can represent a zero-mean WSS process with finite variance in a way similar to the vector representation of a suare-integrable function. Also, strictly speaking, the AWGN does not have finite energy and hence does not have such an expansion. However, since the AWGN model is an approximation to the bandpass additive Gaussian noise, we abuse the mathematics a little bit and assume that the vector representation for the AWGN we consider here is valid. 1.

3 It can be shown (see Homework 1) that rn =[r 1 ;r ;:::;r N ] T is a sufficient statistic for determining which signal is being sent, i.e., determining the value of m. Hence, we only need to deal with vectors of finite dimension. By rewriting (1.) with these finite dimensional vectors, we have rn = smn + nn; (1.6) where smn and nn are the finite truncations of sm and n, respectively. We also note that nn is a zero mean Gaussian random vector whose covariance matrix is N I. The maximum likelihood (ML) receiver makes a decision (select m f; 1;:::;M 1g) which maximizes the likelihood function defined as the following conditional probability density function: NY 1 p(rn jsmn )= p exp " (r # k s mk ) : (1.7) ßN N k=1 Since logarithm is a monotone increasing function, by taking logarithm of p(rn jsmn ), it is easy to see that the ML receiver picks m f; 1;:::;M 1g such that the suared Euclidean distance between the signal vector smn and the receiver vector rn, d (smn; rn ) = NX k=1 (s mk r k ) ; = s T mn s mn s T mn r N + r T N r N (1.8) is minimized. Moreover, since r T N r N is constant for all values of m, wehave where arg min mf1;;:::;mg d (smn; rn ) = arg max mf1;;:::;mg c(s mn; rn ); (1.9) c(smn; rn ) = s T mn r N 1 st mn s mn; = Z T r(t)s m (t)dt E m =: (1.1) In (1.1), c(smn; rn ) is called the correlation metric between the received signal r(t) and the transmitted signal s m (t), and E m = Z T 1.3 s m (t)dt; (1.11)

4 T ( ) dt r(t) s (t) s (t) T ( ) dt - / ε - / 1 ε1 Select Maximum decision T ( ) dt s M-1 (t) εm-1 - / Figure 1.: ML receiver (correlation receiver) for AWGN channel is the energy of the transmitted signal s m (t). The second euality follows from the orthonormal representation (see Homework 1). In summary, we can implement the ML receiver as in the block diagram shown in Figure 1.. This implementation of the ML receiver is called the correlation receiver. 1. Matched filter receiver In this section, we give another implementation, the matched filter receiver, of the ML receiver developed in Section 1.1. For simplicity, we restrict ourselves to the antipodal binary signaling case, i.e., M = and s (t) = s(t) = s 1 (t). It is easy to show (see Homework 1) that the correlators in 1.4

5 T ( ) dt s(t) t=t Figure 1.3: Matched filter Figure 1. can be replaced by the linear filters and samplers as shown in Figure 1.3. As a result, we can employ these linear (matched) filters to implement the ML receiver in Figure 1.. For the antipodal binary case, it is easy to see that the receiver in Figure 1.4 is euivalent to the correlation receiver in Figure 1.. This form of implementation of the ML receiver is known as the matched filter receiver. The matched filter has the optimal property that it is the linear filter that maximizes the output signal-to-noise ratio (SNR). To establish this property, we replace the matched filter in Figure 1.4 by a general linear filter with impulse response h(t). Our goal is to determine the form of h(t) t=t r(t) s(t-t) Y=Y s + Y n >, decide s (t) <, decide s (t) 1 Figure 1.4: Matched filter receiver for AWGN channel (antipodal binary signaling) 1.5

6 the output SNR defined by where SNR = 4 Y s (1.1) E[Yn ]; Y s = Y n = Z T Z T s(t)h(t t)dt; (1.13) n(t)h(t t)dt: (1.14) First, let us evaluate E[Y n ], E[Y n ] = Z T = N = N Z T Z T Z T Z T Substituting (1.15) back into (1.1), we get E[n(fi )n(t)]h(t fi )h(t t)dfidt ffi(fi t)h(t fi )h(t t)dfidt h (T t)dt: (1.15) hr T s(t)h(t t)dti SNR = = N R T h (T t)dt hr T s(t t)h(t)dt i N R T h (t)dt : (1.16) Now by employing the Cauchy-Schwartz ineuality,wehave SNR» N Z T s (T t)dt = E N ; (1.17) witheuality holdsif and onlyifh(t) =Cs(T t) for some constantc. Therefore, the matched filter s(t t), among all linear filters, maximizes the output SNR. We note that the choice of the constant C is immaterial since it does not affect the value of the SNR. We choose C = 1 in this case. An interesting observation from (1.17) is that the maximum SNR achieved by the matched filter depends only on the energy of the signal waveform, but not on other details. Suppose g 1 (t) and g (t) are suare-integrable, then»z 1 Z 1 Z 1» 1 g 1(t)g (t)dt 1 g 1(t)dt 1 g (t)dt; with euality holds if and only if g 1 (t)=cg (t) for some constant C. 1.6

7 1.3 Signal space representation In Section 1.1, we represent the transmitted signals by vectors of finite dimension. It turns out that this geometric viewpoint greatly facilitates the understanding and analysis of many modulation schemes. Because of this, we study the geometric representation more carefully in this section. Suppose fffi n (t)g N n=1 is an orthonormal basis for the signal space spanned by a set of suareintegrable signal waveforms fs m (t)g M 1 m=. We represent the signal waveforms by a set of M N- dimensional vectors with respect to the basis fffi n (t)g N n=1. More precisely, for m = ; 1;:::;M 1, s m (t) is represented by the N-dimensional vector sm = [s m1 ;s m ;:::;s mn ] T whose n-th coordinate s mn,forn =1; ;:::;N;is given by the inner product of s m (t) and ffi n (t), i.e., s mn =(s m ;ffi n ) 4 = Z 1 1 s m(t)ffi n (t)dt: (1.18) Given the basis, we can uniuely 3 determine the signal s m (t) from the vector sm or vice versa. As a result, the vector representation provides a geometric viewpoint of the signal space. Since we are much more familiar with Euclidean geometry than the suare-integrable function space, this geometric viewpoint allows us to visualize the underlying structure of the signal space easily. There are two important identities which greatly simply the analyses in the following sections: (s m ;s k ) d (s m ;s k ) 4 = 4 = Z 1 Z 1 1 s m(t)s k (t)dt = s T m s k; (1.19) 1 [s m(t) s k (t)] dt = ksm skk ; (1.) for m; k =; 1;:::;M 1. The notation k kdenotes the Euclidean norm of a vector. The first identity states that the inner products in the function space and the vector space are euivalent. The second identity states that the suared distance in the function space is the same as the suared Euclidean distance in the vector space. We see from Section 1.1 that the Gram-Schmidt procedure can be employed to find an orthonormal basis from the signal sets. However, we do not always need to employ the Gram-Schmidt procedure to obtain a convenient basis for a signal set. For example, consider the following signal set of the QPSK scheme: s (t) = p P cos(ßt=t + ß=4)p T (t); 3 There can be many bases for the signal space. Different bases give rise to different vector representations. 1.7

8 s 1 PT/ φ s PT/ φ 1 s s 3 Figure 1.5: QPSK constellation s 1 (t) = s (t) = s 3 (t) = p P cos(ßt=t +3ß=4)p T (t); p P cos(ßt=t +5ß=4)p T (t); p P cos(ßt=t +7ß=4)p T (t); where p T (t) =1for» t<t,andp T (t) =otherwise. By inspection, a simple basis for this signal set is ffi 1 (t) = ffi (t) = Using this basis, the corresponding signal vectors are s = [ s 1 = [ s = [ s 3 = [ s s T cos(ßt=t )p T (t); T sin(ßt=t )p T (t): PT=; PT=; PT=] T ; PT=; PT=; PT=] T ; PT=] T ; PT=] T : The corresponding constellation diagram is drawn in Figure 1.5. We will use this simple orthonormal basis to represent the signal spaces of all uadrature modulation schemes. 1.8

9 1.4 ML receiver error analysis In this section, we analyze the performance of the ML receiver by evaluating the symbol error probability. We begin by defining what a symbol error is. We say that a symbol error event occurs when the decision made by the receiver is different from the transmitted symbol. For m =; 1;:::;M 1, let P sjm denotes the conditional symbol error probability given that s m (t) is being transmitted, and P m denotes that probability that the transmitter sends s m (t). Then the average symbol error probability, P s,isgivenby P s = M 1 X m= P sjm P m : (1.1) For simplicity, we assume all the signals are eually likely to be transmitted, i.e., P m = 1=M, for m =; 1;:::;M 1. Then the problem reduces to evaluating P sjm,form =; 1;:::;M 1. We start by working through the simple cases of BPSK and QPSK, for which exact symbol error probabilities can be found. In general, it is often very hard to obtain the exact symbol error probabilities. Therefore, we introduce the method of union bound to upper-bound the symbol error probability BPSK For the case of BPSK (binary antipodal signaling), the matched filter receiver in Section 1. is the ML receiver. The receiver compares the sampled output Y of the matched filter to the threshold zero. If Y >, the receiver decides that s (t) =s(t) is sent. Otherwise, it decides that s 1 (t) = s(t) is sent. From (1.14) and (1.15), we know that the noise sample Y n is a zero mean Gaussian random variable with variance ff Y n = N Z T s (t)dt = EN : Suppose s (t) is being sent, then Y is a Gaussian random variable with mean E and variance ff Y n. From the decision rule stated above, the receiver makes an error when Y». Hence, P sj = Pr(Y» js (t)sent) = 1 p ßff Yn Z 1 exp = Q(E=ff Yn )=Q(p SNR) = Q( # (x E) " dx ffy n E=N ); (1.) 1.9

10 R 1 s 1 R R s s s 5 R 3 R 5 s 4 s 3 R 4 Figure 1.6: Voronoi diagram where Q(x) = 1 p ß Z 1 With the same argument, we can show that P sj1 = P sj = Q( x exp( u =)du: (1.3) E=N ). Therefore, P s = Q( E=N ) General case (a geometric approach) Now assume that we employ M-ary signaling, i.e., the transmitter sends a signal out from the set fs m (t)g M 1 m=. Using the vector representation in Section 1.1, we know that the ML receiver decides that the m-th signal is sent when the Euclidean distance d(rn; smn ) is the smallest among all the M signal vectors. If we draw the signal vectors as points in the constellation diagram as shown in Figure 1.6, the geometric meaning of the ML decision rule is that the signal smn closest to the receiver vector rn is selected. Euivalently, we can construct a decision region (based on the minimum distance principle) for each of the signal point in the constellation diagram, and decide a specific signal point is sent if the received vector rn falls into the corresponding decision region. A diagram showing the signal points and their corresponding decision regions is known as the Voronoi diagram of a modulation scheme. 1.1

11 Now we have all the tools needed to calculate the symbol error probability of a general M-ary modulation with the ML receiver. Suppose s m (t), for some m f; 1;:::;M 1g, is being sent, and let R m denotes the decision region for s m (t). We make an error if the received vector rn falls outside R m. Therefore, the conditional symbol error probability given that s m (t) is sent, P sjm = Pr(rN < N nr m js m (t) sent) = Z = 1 Z < N nr m p(rnjsmn )drn R m p(rn jsmn )drn; (1.4) where p(rn jsmn ) is given in (1.7). Although the expression in (1.4) looks simple, it is generally difficult to construct the Voronoi diagram and evaluate the integral in (1.4). However, for some special cases closed form solutions can be found. The first special case we consider is the binary signaling case (M =, N» ). It is intuitive that the decision regions for the signal points s and s 1 are separated by the hyperplane half-way between the signal points and perpendicular to the line joining the two signal points. The next step is to evaluate the integral in (1.4). Suppose s (t) is being sent, we know that P sj = = Z p(rn js N )drn R 1 Z 1 pßff n exp " (x d(s N; s 1N )=) 1 ffn = Q(d(s N ; s 1N )=ff n ); (1.5) where ff n = N = is the variance of an element of the noise vector nn. We note that the second euality in (1.5) above is obtained by a change of variable which corresponds to a suitable rotation and translation of the axis. Clearly, P sj1 can be calculated in the same way. Thus P s = Q(d(s N ; s 1N )=ff n ). Moreover, it can be seen that the result in (1.5) reduces to (1.) for BPSK (antipodal binary signaling). The next special case we consider is the QPSK example given in Section 1.3 (M =4, N =). It is again obvious that the decision region for a signal point is the uadrant in which the signal point is located. Suppose s (t) is being sent, then P sj = 1 Z R p(r js )dr 1.11 #

12 Z 1 Z 1 = 1 = 1 Q ( 1 ßff n exp 4 (r 1 PT=N )=Q( PT=) +(r PT=) 5 dr1 dr ffn PT=N ) Q ( 3 PT=N ): (1.6) Similarly, we have P sj1 = P sj = P sj3 = P sj and P s =1 Q ( PT=N ) Union bound When the exact symbol error probability is too difficult to evaluate, we resort to bounds and approximations. One of such methods is the union bound. Suppose s (t) is being transmitted, we know from Section 1.4. that P sj = Pr» M 1 X m=1 " M 1 [ m=1 fd(rn; smn ) <d(rn; s N )g fi # fi fis N Pr [fd(rn; smn ) <d(rn; s N )gjs N ] : (1.7) We notice that the event fd(rn; smn ) < d(rn; s N )g in (1.7) is exactly the same as the error event as if there were only two signals, s (t) and s m (t) (m 1), in the signal set. The probability of this event has been calculated in (1.5). Hence, we obtain the union bound of the conditional symbol error probability as P sj» M 1 X m=1 Q(d(s N ; smn )=ff n ); (1.8) where ff n = N =. Similarly, we can find union bounds for the conditional error probabilities given that other signals are sent. By averaging over all the signals, we obtain the union bound for the average symbol error probability as P s» 1 M M 1 X m= M 1 X n= n6=m Q(d(smN; snn )=ff n ): (1.9) As an illustration, we work out the union bound for the symbol error probability for the QPSK example in Section 1.3. From (1.8), we have P sj» Q(d(s N ; s 1N )=ff n )+Q(d(s N ; s N )=ff n )+Q(d(s N ; s 3N )=ff n ) = Q( PT=N )+Q( PT=N )+Q( PT=N ): (1.3) 1.1

13 By symmetry, we have P s» Q( PT=N )+Q( PT=N ); (1.31) which is slightly larger than the exact symbol error probability given in (1.6). 1.5 Complex envelope Very often in a communication system, we do not transmit the lowpass baseband signal directly. Instead, we mix the baseband signal with a carrier up to a certain freuency, which matches the electromagnetic propagation characteristic of the channel. As a result, the actual transmitted signal is a bandpass signal. In this section, we introduce the concept of complex envelope which provides a convenient way to represent bandpass signals Narrowband signal Suppose s(t) is a (real-valued) bandpass signal with most of its freuency content concentrated in a narrow band in the vicinity of a center freuency! c. A sufficient condition is that the Fourier transform of s(t) satisfies S(!) =for j!j! c. We refer to this condition as the narrowband assumption. For a bandpass signal s(t) satisfying the narrowband assumption stated above, it can be shown [1] that s(t) can be represented by an in-phase component x(t) and a uadrature component y(t), s(t) = x(t) cos(! c t) y(t) sin(! c t) ψ e j! ct + e j! ct = x(t) ψ x(t) +jy(t) =!! e j!ct + Now we define the complex envelope ~s(t) of the signal s(t) as ψ e j! ct e j!ct y(t) ψ x(t) jy(t) j!! e j!ct : (1.3) ~s(t) 4 = x(t) +jy(t): (1.33) Then from (1.3), we have s(t) =~s(t)e j!ct =+~s Λ (t)e j!ct ==Re[~s(t)e j!ct ]: (1.34) 1.13

14 Taking Fourier transform on both sides of (1.34), we get S(!) = ~ S(!! c )= + ~ S Λ (!! c )=; (1.35) where S(!) ~ is the Fourier transform of the complex envelope ~s(t). Using (1.34) and (1.35), we can reconstruct the real-valued bandpass signal s(t) back from its complex envelope ~s(t). The remaining uestion is how to obtain the complex envelope ~s(t) from the signal s(t). Ifs(t) is given in the form of (1.3), then we can simply set the complex envelope as ~s(t) =x(t)+jy(t). If only the Fourier transform S(!) ofs(t) isgiven (or s(t) isnot inthe convenientformof (1.3)), morework is needed. First, let us notice that S(!) ~ =for j!j! c. We can conclude this fact easily from (1.35). Based on our narrowband assumption, S(!) = for j!j! c. From (1.35), we know that S(!) is the sum of two shifted (and scaled) versions of S(!). ~ If S(!) ~ does not vanish outside (! c ;! c ), S(!) cannot vanish outside (! c ;! c ). This, of course, contradicts the narrowband assumption. Next, we shift S(!) ~ to the left by! c rad./s in (1.35), S(! +! c )= ~ S(!)=+ ~ S Λ (!! c )=: (1.36) We notice ~ S Λ (!! c ) is nonzero only on the interval ( 3! c ;! c ). Therefore, ~S(!) =L!c [S(! +! c )]; (1.37) where L!c [ ] is the ideal lowpass filter (with bandwidth! c rad./s) operator, which removes all the freuency components outside the band (! c ;! c ). Finally, we can take the inverse Fourier transform of L!c [S(! +! c )] to get ~s(t). Pictorially, we take the positive freuency part of S(!) and shift it down to baseband to obtain S(!). ~ This is the reason why the complex envelope ~s(t) is sometimes called the lowpass euivalent signal of s(t). Based on (1.37), we can easily construct a circuit to convert a real-valued signal to its complex envelope. To do so, we start by rewriting (1.37) in the time domain, ~s(t) = L!c [s(t)e j!ct ] = L!c [s(t) cos(! c t) js(t) sin(! c t)] = L!c [s(t) cos(! c t)] jl!c [s(t) sin(! c t)]: (1.38) 1.14

15 L w c s(t) cos(w c t) ~ s(t) L wc sin(w t) -j c Figure 1.7: Complex envelope conversion circuit We note that the third euality in (1.38) is due to the linearity of the lowpass filter operator L!c.Now it is obvious that we can use the circuit in Figure 1.7 to convert a real-valued signal to its complex envelope Bandpass filter We can use the complex envelope in the previous section to represent the impulse response h(t) of a bandpass filter given that h(t) satisfies the narrowband assumption stated before. Hence, if h(t) ~ is the complex envelope of h(t), then h(t) =Re[ h(t)e ~ j!ct ]: (1.39) Now, if a bandpass signal (satisfying the narrowband assumption) s i (t) is the input to the bandpass filter h(t), then the output from the filter s o (t) also satisfies the narrowband assumption and s o (t) =h(t) Λ s i (t); (1.4) where Λ denotes the convolution operator. In the freuency domain, S o (!) =H(!)S i (!); (1.41) 1.15

16 where S o (!), H(!), ands i (!) are the Fourier transforms of s o (t), h(t), ands i (t), respectively. From (1.37), the Fourier transform of the complex envelope, ~s o (t), ofs o (t) is given by ~S o (!) = L!c [S o (! +! c )] = L!c [H(! +! c )S i (! +! c )] = L!c [H(! +! c )]L!c [S i (! +! c )] = 1 ~H(!) ~ S i (!); (1.4) where ~ H(!) and ~ S i (!) are the Fourier transforms of the complex envelopes, ~ h(t) and ~s i (t), ofh(t) and s i (t), respectively. The third euality in (1.4) is due to the fact that both h(t) and s i (t) share the same passband. By taking inverse Fourier transform on both sides of (1.4), we obtain ~s o (t) = 1 ~ h(t) Λ ~s i (t): (1.43) Hence, we can convolute the complex envelopes of h(t) and s o (t) and then convert the result back to obtain the output bandpass signal Narrowband process Suppose n(t) is a wide-sense stationary (WSS) process with zero mean and power spectral density Φ n (!). If Φ n (!) satisfies the narrowband assumption, then n(t) is called a narrowband process. It turns out [] that n(t) can also be written as n(t) =n x (t) cos(! c t) n y (t) sin(! c t); (1.44) where n x (t) and n y (t) are zero-mean jointly WSS processes. Moreover, if n(t) is Gaussian, n x (t) and n y (t) are jointly Gaussian. By employing the stationarity of the random processes involved, we can show (see Homework 1) that R nx (fi ) = R ny (fi ); (1.45) R nxny (fi ) = R nynx (fi ); (1.46) R n (fi ) = R nx (fi ) cos(! c fi ) R nxny (fi ) sin(! c fi ); (1.47) 1.16

17 where R n (fi ) = 4 E[n(t)n(t + fi )] is the autocorrelation function of the random process n(t), R nx (fi ) = 4 E[n x (t)n x (t + fi )] and R ny (fi ) = 4 E[n y (t)n y (t + fi )] are, respectively, the autocorrelation functions of the processes n x (t) and n y (t), andr nxny (fi ) = 4 E[n x (t)n y (t + fi )] and R nynx (fi ) = 4 E[n y (t)n x (t + fi )] are the cross-correlation functions. Now, let us define the complex envelope ~n(t) of the random process n(t), ~n(t) = 4 n x (t) +jn y (t): (1.48) Obviously, ~n(t) is a zero-mean WSS complex random process with autocorrelation function R ~n (fi ) 4 = 1 E[~nΛ (t)~n(t + fi )] = 1 E[(n x(t) jn y (t))(n x (t + fi )+jn y (t + fi ))] = 1 R n x (fi )+ 1 R n y (fi )+ j R n xn y (fi ) j R n yn x (fi ) = R nx (fi )+jr nxny (fi ): (1.49) Now compare (1.47) with (1.3) and (1.49) with (1.33). If we treat the autocorrelation function R n (fi ) as a bandpass signal (by definition, it satisfies the narrowband assumption since n(t) is a narrowband process), then R ~n (fi ) is its complex envelope. Hence, we can use the results in Section to convert between R n (fi ) and R ~n (fi ). A commonexampleof narrowbandprocess is thebandpassadditivegaussiannoisen(t) with zero mean and power spectral density Φ n (!) = N = for j!j <! c and Φ n (!) = otherwise. Since Φ n (!) satisfies the narrowband assumption, n(t) can be written as n(t) =n x (t) cos(! c t) n y (t) sin(! c t); (1.5) where n x (t) and n y (t) are zero-mean jointly WSS Gaussian processes. The complex envelope of n(t) is given by ~n(t) =n x (t) +jn y (t): (1.51) Using the result above and (1.37), the power spectral density Φ ~n (!) of the complex envelope ~n(t) is given by Φ ~n (!) = L!c [Φ n (! +! c )] = 8 >< >: N if j!j <! c ; otherwise: (1.5) 1.17

18 Taking inverse Fourier transform, we get where! c sin(! c fi ) R ~n (fi )=N f c ; (1.53)! c fi = ßf c. Since R ~n (fi ) is real, R nxn y (fi ) = and hence the processes n x (t) and n y (t) are uncorrelated. Moreover, we have R nx (fi ) = R ny (fi ) = R ~n (fi ). For the case where bandpass transmitted signals are sent through a channel corrupted by n(t) and the bandwidths of the transmitted signals are much smaller than the carrier freuency! c, we approximate R ~n (fi ) in (1.53) by N ffi(fi ). This means that the lowpass euivalent of the additive bandpass Gaussian noise looks white to the lowpass euivalents of the transmitted signals. In this way, we are back to the communication model of transmitting baseband signals over an AWGN channel as in Section 1.1. Of course, all the signal are complex instead of real now. By using the same method in Section1.1, we can develop (see Homework 1) the ML receiver for the complex baseband communication system. 1.6 Noncoherent receiver The ML receiver is developed in Section 1.1 based on the assumption that the channel does nothing to the transmitted signal except adding the AWGN to it. This model is obviously too simple to model any real life communication channel. As we mentioned before, since most communication systems transmit bandpass signals instead of baseband ones, we focus on this kind of signals and use the complex envelopes to represent them here. Again, we consider the simple case of a non-dispersive channel, for which we can model the received signal as r(t) =Ae j s m (t) +n(t); 4 (1.54) where A> represents the channel gain (attenuation), represents the carrier phase shift due to propagation delay, local oscillator mismatch, and etc.,andn(t) is the complex AWGN with autocorrelation function R n (fi ) = N ffi(fi ). Suppose the receiver knows the value of 5, the problem reduces to the one in Section 1.1. Hence we can use the correlation receiver in Figure 1. (or its complex euivalent) 4 From now on, we drop the symbol for complex envelopes. 5 The value of A is not needed when the signals have eual energies 1.18

19 T ( ) dt or r(t) s * (t) s 1* (t) T ( ) dt or Select Maximum decision s * M-1 (t) T ( ) dt or Figure 1.8: Envelope / suare-law receiver for M-ary orthogonal signals to detect the received signal r(t). Generally, receivers that make use of the phase information are referred to as coherent receivers. Therefore, the correlation receiver in Figure 1. and the matched filter receiver in Figure 1.4 are coherent receivers. For coherent reception, we need to estimate the carrier phase. This estimation can sometimes be hard to perform, and inaccurate estimation of the carrier phase will significantly degrade the performance of the coherent ML receiver. One alternative to coherent reception is to avoid using the phase information. To do so, we model the carrier phase as a random variable uniformly distributed on [; ß). Following steps similar to those in Section 1., we can develop the ML receiver for this case. The resulting receiver is known as the noncoherent ML receiver. For the case where the transmitted signals fs m (t)g M 1 m= have eual energies, the ML receiver assumes the simple form [1] shown in Figure 1.8. This receiver is usually referred to as the envelope receiver or the suare-law receiver 1.19

20 depending on whether the envelope or the suare-law detecting device is employed. It is difficult to evaluate the symbol error probability for a general M-ary signal set received by the noncoherent ML receiver. For the special case of eual-energy binary orthogonal signals, we state that the average symbol error probability (assuming eual a priori probabilities) is given by [1] P s = 1 e E=N ; (1.55) where E is the signal energy. 1.7 Power spectrum In all the previous sections, we assume that a single time-limited signal (pulse) is sent. In this section, we consider a more realistic model in which a train of pulses are transmitted. For simplicity, we ignore the white noise and assume that the (complex envelope of the) received signal is given by s(t) = x(t) +jy(t); (1.56) x(t) = y(t) = 1X k= 1 1X k= 1 a k ψ x (t kt s ); (1.57) b k ψ y (t kt s ); (1.58) where a k s are independent identically distributed (iid) random variables with mean zero and variance A,andb k s are also iid random variables with mean zero and variance B. Moreover, we assume that the two data streams fa k g 1 k= 1 and fb kg 1 k= 1 are independent. In above, can be interpreted as the propagation delay, and ψ x (t) and ψ y (t) are the pulses for the in-phase and uadrature channels, respectively. We notice thats(t) is a zero-mean random process. This model almost covers all practical uadrature modulation schemes. Our objective is to evaluate the autocorrelation function of s(t). First, let us model as a random variable which is uniformly distributed on [;T s ), and is independent to both fa k g 1 k= 1 and fb kg 1 k= 1. Then the autocorrelation function of s(t) is given by R s (t; t + fi ) 4 = 1 E[sΛ (t)s(t + fi )] = 1 = 1 fe[x(t)x(t + fi )] je[x(t)y(t + fi )] je[y(t)x(t + fi )] + E[y(t)y(t + fi )]g fe[x(t)x(t + fi )] + E[y(t)y(t + fi )]g : (1.59) 1.

21 The last euality in (1.59) follows from the fact that the two data streams consist of zero-mean independent random variables. Now, it suffices to evaluate E[x(t)x(t + fi )], E[x(t)x(t + fi )] = = 1X 1X k= 1 l= 1 1X k= 1 = A T s E[a k a l ]E[ψ x (t kt s )ψ x (t + fi lt s )] Z A 1 Ts ψ x (t kt s )ψ x (t + fi kt s )d T s 1X k= 1 Z (k+1)ts kt s ψ x (t )ψ x (t + fi )d Z 1 = A ψ x (t )ψ x (t + fi )d T s 1 Z 1 = A ψ x ( )ψ x (fi )d T s 1 = A ψ x ( fi ) Λ ψ x (fi ): (1.6) T s Similarly, we have E[y(t)y(t + fi )] = B ψ y ( fi ) Λ ψ y (fi ): (1.61) T s Therefore, the process s(t) is WSS and R s (fi )=R s (t; t + fi )= 1 h A ψ x ( fi ) Λ ψ x (fi )+B ψ y ( fi ) Λ ψ y (fi ) T s The power spectral density (power spectrum) of s(t) is given by i : (1.6) Φ s (!) = 1 T s h A jψ x (!)j + B jψ y (!)j i ; (1.63) where Ψ x (!) and Ψ y (!) are the Fourier transforms of ψ x (t) and ψ y (t), respectively. For example, we consider the BPSK scheme where ψ x (t) = p Ts (t) and ψ y (t) =. We consider two cases: T s = T and T s = T=1. In both cases, we let A =. For the first case, the power spectrum is Φ s (!) =T sin (!T=) : (1.64) (!T=) For the second case, the power spectrum is T sin (!T=) Φ s (!) = : (1.65) 1 (!T=) The two spectra are plotted in Figure 1.9 for comparison. From Figure 1.9, we observe that if we decrease the pulse duration, we will obtain a wider and lower power spectrum. This observation forms the basis for spread spectrum communications. 1.1

22 1 T s =T/1 T s =T 1 1 power spectral density ω (π/t Hz) Figure 1.9: Power spectra of BPSK schemes: T s = T and T s = T=1 1.

23 1.8 References [1] J. G. Proakis, Digital Communications, 3rd Ed., McGraw-Hill, Inc., [] W. B. Davenport and W. L. Root, An Introduction to the Theory of Random Signals and Noise, McGraw-Hill, Inc.,